Primes Appearing in Prime Tower Factorization

TL;DR

This paper investigates the probability of a fixed prime appearing in a recursively defined integer factorization, deriving infinite product formulas and algorithms for precise probability bounds.

Contribution

It introduces a novel probability question in number theory and develops methods to compute bounds on these probabilities despite lacking closed-form solutions.

Findings

Derived convergent infinite product formulas for the probability

Developed algorithms to compute arbitrarily close bounds

Provided insights into prime appearances in recursive factorizations

Abstract

We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent infinite products for this probability, which we are unable to simplify to obtain closed form solutions. However, we are able to implement these formulas in the development of algorithms to obtain arbitrarily close rigorous bounds on the probabilities in question.